Friedberg numbering in fragments of Peano Arithmetic and $\alpha$-recursion theory

Abstract

In this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchy $L_\alpha$, where $\alpha$ is $\Sigma_1$ admissible. We prove that

(1) Over $P^-+B\Sigma_2$, the existence of a Friedberg numbering is equivalent to $I\Sigma_2$, and

(2) For $L_\alpha$, there is a Friedberg numbering if and only if the tame $\Sigma_2$ projectum of $\alpha$ equals the $\Sigma_2$ cofinality of $\alpha$.